77.1.51 problem 70 (page 112)
Internal
problem
ID
[17941]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
70
(page
112)
Date
solved
:
Tuesday, January 28, 2025 at 11:14:15 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-x^{3} \left (1-y^{\prime }\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.128 (sec). Leaf size: 190
dsolve(diff(y(x),x)^3-x^3*(1-diff(y(x),x))=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (\int \frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {12 x^{3}+81}\right )^{{2}/{3}}+12 i \sqrt {3}\, x -\left (108+12 \sqrt {12 x^{3}+81}\right )^{{2}/{3}}+12 x \right ) x}{\left (108+12 \sqrt {12 x^{3}+81}\right )^{{1}/{3}}}d x \right )}{12}+c_{1} \\
y &= -\frac {\left (\int \frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {12 x^{3}+81}\right )^{{2}/{3}}+12 i \sqrt {3}\, x +\left (108+12 \sqrt {12 x^{3}+81}\right )^{{2}/{3}}-12 x \right ) x}{\left (108+12 \sqrt {12 x^{3}+81}\right )^{{1}/{3}}}d x \right )}{12}+c_{1} \\
y &= \frac {\left (\int \frac {\left (\left (108+12 \sqrt {12 x^{3}+81}\right )^{{2}/{3}}-12 x \right ) x}{\left (108+12 \sqrt {12 x^{3}+81}\right )^{{1}/{3}}}d x \right )}{6}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 175.927 (sec). Leaf size: 368
DSolve[D[y[x],x]^3-x^3*(1-D[y[x],x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \int _1^x\frac {\sqrt [3]{2} \left (9 K[1]^3+\sqrt {3} \sqrt {K[1]^6 \left (4 K[1]^3+27\right )}\right )^{2/3}-2 \sqrt [3]{3} K[1]^3}{6^{2/3} \sqrt [3]{9 K[1]^3+\sqrt {3} \sqrt {K[1]^6 \left (4 K[1]^3+27\right )}}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) K[2]^3+i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (18 K[2]^3+2 \sqrt {3} \sqrt {K[2]^6 \left (4 K[2]^3+27\right )}\right )^{2/3}}{12 \sqrt [3]{9 K[2]^3+\sqrt {3} \sqrt {K[2]^6 \left (4 K[2]^3+27\right )}}}dK[2]+c_1 \\
y(x)\to \int _1^x\frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) K[3]^3+\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (18 K[3]^3+2 \sqrt {3} \sqrt {K[3]^6 \left (4 K[3]^3+27\right )}\right )^{2/3}}{12 \sqrt [3]{9 K[3]^3+\sqrt {3} \sqrt {K[3]^6 \left (4 K[3]^3+27\right )}}}dK[3]+c_1 \\
\end{align*}